| L(s) = 1 | − 3-s − 4·7-s + 9-s − 13-s − 2·17-s + 4·21-s − 27-s − 2·29-s − 4·31-s − 6·37-s + 39-s − 6·41-s − 4·43-s + 4·47-s + 9·49-s + 2·51-s + 10·53-s − 2·61-s − 4·63-s − 8·67-s + 4·71-s + 6·73-s − 8·79-s + 81-s − 8·83-s + 2·87-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.485·17-s + 0.872·21-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.986·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s + 0.583·47-s + 9/7·49-s + 0.280·51-s + 1.37·53-s − 0.256·61-s − 0.503·63-s − 0.977·67-s + 0.474·71-s + 0.702·73-s − 0.900·79-s + 1/9·81-s − 0.878·83-s + 0.214·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6239469990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6239469990\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.67159140531585010270478255658, −6.91171614802038001637693172711, −6.61774978948423487153263219027, −5.75647329558749384709319033834, −5.23707133240235510872706130496, −4.23385175769989709645609166290, −3.55395396206400240215335899903, −2.77379309800769604842954636721, −1.74471435805599907598887883889, −0.39126205838128547205640568104,
0.39126205838128547205640568104, 1.74471435805599907598887883889, 2.77379309800769604842954636721, 3.55395396206400240215335899903, 4.23385175769989709645609166290, 5.23707133240235510872706130496, 5.75647329558749384709319033834, 6.61774978948423487153263219027, 6.91171614802038001637693172711, 7.67159140531585010270478255658
